A036667 Numbers of the form 2^i*3^j, i+j even.

1, 4, 6, 9, 16, 24, 36, 54, 64, 81, 96, 144, 216, 256, 324, 384, 486, 576, 729, 864, 1024, 1296, 1536, 1944, 2304, 2916, 3456, 4096, 4374, 5184, 6144, 6561, 7776, 9216, 11664, 13824, 16384, 17496, 20736, 24576, 26244, 31104, 36864, 39366

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A257999 with respect to A003586.

Intersection of A028260 and A003586.

Cf. A025620 (subsequence), A069352, A022328, A022329.

Programs

Haskell

a036667 n = a036667_list !! (n-1)
a036667_list = filter (even . flip mod 2 . a001222) a003586_list
-- Reinhard Zumkeller, May 16 2015

Mathematica

max = 40000;
Reap[Do[k = 2^i 3^j; If[k <= max && EvenQ[i+j], Sow[k]], {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[3, max] // Ceiling}]][[2, 1]] // Union (* Jean-François Alcover, Aug 04 2018 *)

Python

from sympy import integer_log
def A036667(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length()+(i&1^1)>>1 for i in range(integer_log(x, 3)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

Formula

A069352(a(n)) mod 2 = 0. - Reinhard Zumkeller, May 16 2015

Sum_{n>=1} 1/a(n) = 7/4. - Amiram Eldar, Feb 18 2021

Extensions

Offset corrected by Reinhard Zumkeller, May 16 2015